A 2-order additive fuzzy measure identification method based on hesitant fuzzy linguistic interaction degree and its application in credit assessment

To reflect both fuzziness and hesitation in the evaluation of interactivity between attributes in the identification process of 2-order additive fuzzy measure, this work uses the hesitant fuzzy linguistic term set (HFLTS) to describe and depict the interactivity between attributes. Firstly, the interactivity between attributes is defined by the supermodular game theory. According to this definition, a linguistic term set is established to characterize the interactivity between attributes. Under the linguistic term set, the experts employ linguistic expressions generated by context-free grammar to qualitatively describe the interactivity between attributes. Secondly, through the conversion function, the linguistic expressions are transformed into the hesitant fuzzy linguistic term sets (HFLTSs). The individual evaluation results of all experts were further aggregated with the defined hesitant fuzzy linguistic weighted power average operator (HFLWPA). Thirdly, based on the standard Euclidean distance formula of the hesitant fuzzy linguistic elements (HFLEs), the hesitant fuzzy linguistic interaction degree (HFLID) between attributes is defined and calculated by constructing a piecewise function. As a result, a 2-order additive fuzzy measure identification method based on HFLID is proposed. Based on the proposed method, using the Choquet fuzzy integral as nonlinear integration operator, a multi-attribute decision making (MADM) process is then presented. Taking the credit assessment of the big data listed companies in China as an application example, the analysis results of application example prove the feasibility and effectiveness of the proposed method. This work successfully reflects both the fuzziness and hesitation in evaluating the interactivity between attributes in the identification process of 2-order additive fuzzy measure, enriches the theoretical framework of 2-order additive fuzzy measure, and expands the applicability and methodology of 2-order additive fuzzy measure in multi-attribute decision making.

www.nature.com/scientificreports/ to general fuzzy measure (k = n).Among them, the 2-order additive fuzzy measure only needs to determine n(n + 1)/2 parameters, involving the relative importance of attributes and the interaction between attributes, which solves the contradiction between complexity and performance ability well, so it has been widely used 7 .
However, due to the difficulty in explaining and understanding the interaction between attributes 8 and the uncertainty of decision makers' cognition, decision makers often cannot give accurate quantitative evaluations of interactivity between attributes, which is often fuzzy and hesitant.The existing 2-order additive fuzzy measure identification methods mainly use subjective methods [9][10][11][12][13][14][15][16] and objective methods 7,[17][18][19][20][21][22] to describe and depict the interactivity between attributes.Compared with the objective methods, the subjective methods are more explanatory, so they have been applied more widely.However, in the existing subjective methods, most methods cannot reflect the fuzziness of decision-making, while a few methods, although able to reflect the fuzziness of decision-making, cannot reflect the hesitation of decision-making.Therefore, it is urgent to construct an evaluation method for the interactivity between attributes that can reflect both fuzziness and hesitation.
In 23 , Rodríguez et al. introduced the concept of a hesitant fuzzy linguistic term set (HFLTS) to provide a linguistic and computational basis for increasing the flexibility and richness of linguistic elicitation based on the fuzzy linguistic approach and the use of context-free grammar to support the elicitation of linguistic information by experts in hesitant situations under qualitative settings.Hesitant fuzzy linguistic information is based on the linguistic expressions given by people; therefore, it is closer to people's thinking and cognition; and can flexibly and comprehensively reflect the real preferences of decision-makers 24 .HFLTS provides a new and powerful tool to characterize experts' qualitative decision information and has thus been successfully applied in the field of uncertain MADM [25][26][27][28] .
Based on this observation, to reflect both fuzziness and hesitation in the evaluation of interactivity between attributes, the present work uses the HFLTS to describe and depict the interactivity between attributes.As a result, a 2-order additive fuzzy measure identification method based on HFLID is proposed.The marginal contributions of this paper may be as follows: 1) The interactivity between attributes is defined by the supermodular game theory.According to this definition, a linguistic term set is established to characterize the interactivity between attributes.Under the linguistic term set, the experts employ the linguistic expressions generated by the context-free grammar to qualitatively describe the interactivity between attributes.(2) Through the conversion function, the linguistic expressions are transformed into HFLTSs.The individual evaluation results of all experts are further aggregated with the defined hesitant fuzzy linguistic weighted power average operator (HFLWPA).
(3) Based on the standard Euclidean distance formula of the hesitant fuzzy linguistic elements (HFLEs), the hesitant fuzzy linguistic interaction degree (HFLID) between attributes is defined and calculated by constructing a piecewise function.
Based on the proposed method, using the Choquet fuzzy integral as a nonlinear integration operator 29,30 , a MADM process is further presented and applied to the credit assessment of big data enterprises.The remaining part of this paper is structured as follows: "Preparatory knowledge" introduces the preparatory knowledge employed in this study."The proposed method" proposes the 2-order additive fuzzy measure identification method based on HFLID."A MADM process based on the proposed method" presents the MADM process based on the proposed method."Application Example" describes the application example analysis results."Discussion" discusses the results obtained and "Conclusion" concludes this paper.

Preparatory knowledge
This section introduces the related definitions of HFLTS, 2-order additive fuzzy measure and Choquet fuzzy integral reported in the literature.This is the basis of "The proposed method" and "A MADM process based on the proposed method".

Related definitions of HFLTS
Definition 1 23,31 : Let X = {x 1 , x 2 , • • • , x n } be a universe of discourse, and    , τ be a linguistic term set.A hesitant fuzzy linguistic term set (HFLTS) in X is an object having the form where h S (x i ) is the set of elements in S, called hesitant fuzzy linguistic element (HFLE), it can be expressed as Definition 2 24,31 : Let G H be a context-free grammar, and S = s The elements of G H = (V N , V T , I, P) are defined as follows: V N = {primary term, composite term, unary relation, binary relation, conjunction}; V T = {"less than", "more than", "at least", "at most", "between", "and", " s −τ ", • • • , " s −1 ", " s 0 ", " s 1 ", • • • , " s τ "}; I ∈ V N ; P = {I refers to the primary term or composite term; the primary term refers to " s −τ ", • • • , " s −1 ", " s 0 ", " s 1 ", • • • , " s τ "; the composite term refers to unary relation + primary term, or binary relation + primary term + conjunction + primary term; the unary relation refers to "less than" or "more than" or "at least" or "at most"; the binary relation refers to "between"; the conjunction refers to "and"}.
b e a l i n g u i s t i c t e r m s e t , h S = s η l s η l ∈ S, l = 1, 2, • • • , L is a HFLE defined on S, then the mean value of h S is defined as and the variance of h S is defined as T h e n , t h e b i n a r y r e l a t i o n s h i p b e t w e e n h i S = s η i l For convenience of calculation, referring to 33 , the h S with fewer elements can be expanded by adding element until meeting the need of calculation, where s + η l and s − η l is the largest and smallest element in h S respectively, and ζ ∈ [0, 1] is the adjustment parameter.Without loss of generality, ζ = 0.5 is usually taken.(1) sup(h i S , is the power set of X, if the set function g : P(X) → [0, 1] satisfies the following two conditions:
Grabisch 6 proposed the k-order additive fuzzy measure based on pseudo-Boolean function and Möbius transformation.On this basis, the 2-order additive fuzzy measure is then defined as where m i is the Möbius transformation coefficient of which represents the extent of interac- tion between x i and x j .Definition 8 13 : Let X = {x 1 , x 2 , • • • , x n } be a set of attributes, W = {w 1 , w 2 , • • • , w n } is the weight set of X, the Möbius transformation coefficients of x i and x i , x j are respectively where P = i∈X * w i + {i,j}⊂X * ξ ij w i w j is the sum of the importance of all x i and x i , x j , ξ ij is the interaction degree between x i and x j , Definition 9 35 : Let f be a nonnegative function defined on X, F is a σ-algebra composed of subsets of X (when X is finite, F is the power set P(X) of X), g is a fuzzy measure defined on F, then the Choquet fuzzy integral of function f on set X for fuzzy measure g is defined as where )dα is the Riemann integral.When X is a finite set, the elements in X are rearranged as . Let H=(c) fdg , then the Choquet fuzzy integral has the following simpli- fied formula: where

The proposed method
This section uses the HFLTS to describe and depict the interactivity between attributes, and then proposes a 2-order additive fuzzy measure identification method based on HFLID.In addition, the correctness of the proposed method is proved theoretically.
be a set of attributes to compare the alternatives, let C * = {1, 2, • • • , n} be a set of subscripts of attributes.The weight vector of attributes is ) be a set of experts, the weight vector of experts is , where w p ∈ [0, 1] , and t p=1 w p = 1 .Using the HFLTS, the identification process of 2-order additive fuzzy measure based on HFLID is shown in Fig. 1.
Step 1: Establish a linguistic term set S to characterize the interactivity between C i and C j (i ≠ j).
According to the supermodular game theory 36 , the interactivity between attributes is defined as follows: Definition 10 Let C i and C j (i ≠ j) have a partial order relation in the attribute set C, and the supremum C i ∨ C j and infimum C i ∧ C j are in C, then C is called a sub-lattice 36 .Let f be a real-valued function defined on a sub- lattice then there is complementarity between C i and C j (i ≠ j); when f f is a submodular function 36 , then there is repeatability between C i and C j (i ≠ j); in particular, when , then there is independence between C i and C j (i ≠ j). (3) Vol.:(0123456789) to char- acterize the interactivity between C i and C j (i ≠ j), where s 1 , s 2 , • • • , s τ are the linguistic terms describing com- plementarity, the larger β is, the stronger the complementarity is; s −τ , s −τ +1 , • • • , s −1 are the linguistic terms describing repeatability, the smaller β is, the stronger the repeatability is; s 0 is then a linguistic term describing independence, at this point, β= 0.
Step 2: Calculate the individual evaluation result of the expert Under the linguistic term set S, according to Definition 10, every expert employs the linguistic expressions ll generated by the context-free grammar G H (see Definition 2) to evaluate the interactivity between C i and C j (i ≠ j) ( C 2 n pairs in total).Through the transformation function E G H : ll → H ij S (see Definition 3), the linguistic expressions ll are further transformed into the HLFTS ) on the interactivity between C i and C j (i ≠ j) is then given.
Step 3: Calculate the group evaluation result of t experts on the interactivity between C i and C j (i ≠ j).
Based on Definition 6, considering the weights of experts, the hesitant fuzzy linguistic weighted power average operator (HFLWPA) is defined as follows: are t HFLEs defined on S. The weight vector of t HFLEs is (w 1 , w 2 , • • • , w t ) , where w p ∈ [0, 1] , and t p=1 w p = 1 .Let HFLWPA : t → , if then HFLWPA is called the hesitant fuzzy linguistic weighted power average operator (when w p = 1/t , HFLWPA degenerates to HFLPA 34 ), where T(h S , it satisfies the following three conditions: (1) sup(h In 37 , Yager defined different support functions, which lead to different degrees of support.In this paper, we take the support function as follows: sup(h According to Definition 4, after expanding those h ij(p) S ( p = 1, 2, • • • , t ) with fewer elements, using the HFL- WPA to aggregate the individual evaluation results of t experts, the group evaluation result of t experts on the interactivity between C i and C j (i ≠ j) is then obtained as Step 4: Determine the HFLID H( ξij ) between C i and C j (i ≠ j).According to the established linguistic term set S (see step 1), by constructing a piecewise function based on Definition 5, the hesitant fuzzy linguistic interaction degree (HFLID) between attributes is defined as follows: www.nature.com/scientificreports/ Based on Definition 5a piecewise function is constructed as where H( ξij ) is called the hesitant fuzzy linguistic interaction degree (HFLID) between C i and C j (i ≠ j), obviously, If there is complementarity between C i and C j (i ≠ j), then H( ξij ) > 0 , and the larger H( ξij ) , the stronger complementarity.If there is repeatability between C i and C j (i ≠ j), then H( ξij ) < 0 , and the smaller H( ξij ) , the stronger repeatability.If C i and C j (i ≠ j) are independent of each other, then H( ξij ) = 0.
Therefore, according to Definition 12, the HFLID H( ξij ) between C i and C j (i ≠ j) can be determined.
Step 5: Calculate the Möbius transformation coefficients m i and m ij of attributes.According to the weight vector W C = (w 1 , w 2 , • • • , w n ) of attributes and the HFLID H( ξij ) between C i and C j (i ≠ j), the Möbius transformation coefficients m i and m ij of the attributes are calculated by using Eq. ( 4).The calculation formula is as follows: where P = i∈C * w i + {i,j}⊂C * H( ξij )w i w j is the sum of the importance of all C i and C i , C j (i ≠ j).
Step 6: Identify the 2-order additive fuzzy measure g K .
According to the Möbius transformation coefficients m i and m ij of attributes, the 2-order additive fuzzy measure g K are calculated using Eq. ( 3).The calculation formula is as follows: Theorem 1: The fuzzy measure identified by steps 1 to 6 is a 2-order additive fuzzy measure.To prove that the fuzzy measure identified by steps 1 to 6 is a 2-order additive fuzzy measure, it is only necessary to prove that the calculated Möbius transformation coefficients satisfy the following constraints 6 : (1) m(∅) = 0; (2) Proof: (1) m(∅) = 0 , obviously holds.
(2) Because H( ξij ) = H( ξji ) , and n i=1 w i = 1 , P can be further written as Because H( ξij ) ∈ [−1, 1] , and w j ∈ [0, 1] , we have −w j ≤ H( ξij )w j ≤ w j , i = j.Sum the two sides of the above inequality to j, we obtain Multiplying both sides of the above inequality by w i , we obtain Sum the two sides of the above inequality to i, and the following inequality can be given Because P > 0 , and w i ≥ 0 , we can get m i = w i P ≥ 0. ( Thus, the following inequality can be given Hence, we get

A MADM process based on the proposed method
According to the 2-order additive fuzzy measure identification method based on HFLID, this section presents a MADM process.
Taking the Choquet fuzzy integral as a nonlinear integration operator, the MADM process based on the proposed method (see "The proposed method") is as follows: Step 1: Construct the normalized decision matrix X.
According to the types of attributes (including positive type, negative type, neutral type), the decision matrix X is normalized to construct a normalized decision matrix X: where xji is the normalized attribute value of alternative Step 2: Determine the HFLID H( ξij ) between C i and C j (i ≠ j).
Given the weight vector W D of experts, the HFLID H( ξij ) between C i and C j (i ≠ j) can be determined by step 1-4 in "The proposed method".
Step 3: Calculate the Möbius transformation coefficients m i and m ij of attributes.Given the weight vector W C of attributes, according to the HFLID H( ξij ) between C i and C j (i ≠ j), the Möbius transformation coefficients m i and m ij of the attributes are calculated by step 5 in "The proposed method".
Step 4: Identify the 2-order additive fuzzy measure g K .
According to the Möbius transformation coefficients m i and m ij of attributes, the 2-order additive fuzzy measure g K can be further identified by step 6 in "The proposed method".
Step 5: Calculate the Choquet fuzzy integral values and the ranking of alternatives.By reordering the normalized attribute value xji ) from small to large, the xj(i) can be obtained.Substituting the xj(i) and the 2-order additive fuzzy measure g K into Eq.( 5), the Choquet fuzzy integral value H j of the alternative A j can be calculated.Simultaneously, the ranking of alternatives can be given, where the larger H j , the better the alternative A j .

Application Example
This section uses the application example and data from 16 to demonstrate the feasibility and effectiveness of the MADM process based on the proposed method.www.nature.com/scientificreports/

Credit assessment index system and sample data
Considering the characteristics of big data enterprises 38 , and following the principles of selecting indicators, such as scientificalness, objectivity, systematization, functionality, dynamics, relative independence, feasibility (or operability), comparability and so on, a credit assessment index system for big data enterprises was constructed, it included 6 primary indicators and 16 secondary indicators (see Table 1 in 16 ), where the primary indicators included Debt Paying Ability (C 1 ), Operational Capability (C 2 ), Profitability (C 3 ), Growth Capability (C 4 ), Technological Innovation Capability (C 5 ) and Industry Growth (C 6 ).We selected the big data listed companies in the Growth Enterprise Market (GEM) in China -Wangsu Science & Technology Co., Ltd.(300017), Beijing Lanxum Technology Co., Ltd.(300010), and Wuhan Tianyu Information Industry Co., Ltd.(300205) to form a set of alternatives, denoted by A = {A 1 , A 2 , A 3 }, where the alternative A 1 and A 2 belong to the software service industry, and the alternative A 3 belongs to the electronic components industry.The sample data were the section data of 2016.We obtained a total of 48 original data (see Table 2 in 16 ), where the original data of the Number of Invention Patent Applications Announced were from the Tian Yan Cha website, the original data of the Network Attention of Industry were from the Baidu Index website, and the rest of the original data were from the East Money website.

Process and results of credit assessment
Step 1: Construct the normalized decision matrix.
Using the algorithm given in 39 , based on the original data, the weight vector W C of attributes was calculated as 16 W C = (0.1702, 0.1708, 0.1810, 0.1208, 0.2261, 0.1312).
Step 2: Determine the HFLID H( ξij ) between C i and C j (i ≠ j).According to step 1 in "The proposed method", a linguistic term set S was established to characterize the interactivity between C i and C j (i ≠ j), where S = {s −4 = repeatability is extremely strong, s −3 = repeatability is very strong, s −2 = repeatability is strong, s −1 = repeatability is relatively strong, s 0 = independence, s 1 = complementarity is relatively strong, s 2 = complementarity is strong, s 3 = complementarity is very strong, s 4 = complementarity is extremely strong }.
That is to say, the credit status of the alternative A 1 was relatively good, and the credit status of the alternative A 3 was relatively poor.

Comparative analysis
For comparison, we invited the above three experts to use the scoring method 13 , the diamond pairwise comparisons (DPC) method 9 and the intuitionistic fuzzy sets (IFSs) method 16 to determine the interaction degrees between attributes, respectively.Given the weight vector W D = (0.3976, 0.3012, 0.3012) of experts, the scoring interaction degrees 13 , the DPC interaction degrees 9 and the intuitionistic fuzzy interaction degrees 16 between attributes were further obtained, as shown in Table 5.
Furthermore, we replaced the HFLIDs (see Table 2) in step 2 of "Process and results of credit assessment" with the scoring interaction degrees (see Table 5).Ceteris paribus, the Choquet fuzzy integral values of alternatives were calculated as , then the ranking of alternatives was A 1 ≻ A 2 ≻ A 3 .Similarly, we also replaced the HFLIDs (see Table 2) in step 2 of "Process and results of credit assessment" with the DPC interaction degrees (see Table 5).Ceteris paribus, the Choquet fuzzy integral values of alternatives were calculated as , then the ranking of alternatives was A 1 ≻ A 2 ≻ A 3 .In addition, we also replaced the HFLIDs (see Table 2) in step 2 of "Process and results of credit assessment" with the intuitionistic fuzzy interaction degrees (see Table 5).Ceteris paribus, the Choquet fuzzy integral values of alternatives were calculated as , then the ranking of alternatives was A 1 ≻ A 2 ≻ A 3 .that of Method 2 (its value was equal to 0.0355), Method 3 (its value was equal to 0.0449) and Method 4 (its value was equal to 0.0684).According to Eq. ( 4), the variances of m i and m ij increase with the increase of the variance of interaction degrees.From Eq. (3) and Eq. ( 5), we can further see that the variance of g(K) and the variance of H also increase correspondingly.Thus, according to the algorithm of discrimination 42 , the discrimination value becomes larger.This transmission mechanism is as follows: where and σ 2 H represents the variance of ξ ij , m i , m ij , g(K) and H , respectively.Compared with Method 2, Method 3 and Method 4, Method 1 can obtain the slightly lower Choquet fuzzy integral mean value.Since the average value of interaction degrees determined by Method 1 (its value was equal to 0.3023) was higher than that of Method 2 (its value was equal to 0.2087), Method 3 (its value was equal to 0.2490) and Method 4 (its value was equal to 0.2718).According to Eq. ( 4), when the average value of interaction degrees increases, the mean value of P increases, meanwhile, the mean value of m i decreases and the mean value of m ij increases.However, because |m i | ≫ m ij , from Eq. ( 3), we can further see that the mean value of g(K) also decreases correspondingly.Thus, according to Eq. ( 5), the mean value of H becomes smaller.This transmission mechanism is as follows: where ξ ij , P , m i , m ij and g(K) represents the average value of ξ ij , P , m i , m ij and g(K) , respectively.
Both the variance and the average value of interaction degrees determined by Method 1 were higher than that of Method 2, Method 3 and Method 4, the fundamental reasons are as follows: 1) The Method 1 defines the interactivity between attributes by using the supermodular game theory, so that the interaction between attributes is easier to explain and understand.According to this definition, the linguistic term set S is established to characterize the interactivity between attributes.The experts then employ the linguistic expressions generated by the context-free grammar to qualitatively describe the interactivity between attributes under the linguistic term set S, which makes the evaluation of the interactivity between attributes closer to reality.2) In Method 1, through the conversion function, the linguistic expressions are transformed into the HFLTSs.The individual evaluation results of all experts are further aggregated by using the defined HFLWPA.The HFLID between attributes is then defined and calculated, thereby better preserving, and characterizing the experts' evaluation information.3) In Method 2, the experts use scoring method to determine the interaction degree between attributes according to experience and preference.The Method 3 uses diamond diagram to help decision makers intuitively determine the relative importance and interaction coefficient between attributes.These two methods lack both the theoretical definition of the interactivity between attributes and the fuzziness and hesitation of decision-making.In addition, the Method 4 uses the intuitionistic fuzzy sets (IFSs) to describe and depict the interactivity between attributes, however, the IFSs cannot reflect the hesitation of expert decision-making.

Conclusion
The marginal contributions of this paper may be as follows: 1) The interactivity between attributes is defined by the supermodular game theory.According to this definition, a linguistic term set is established to characterize the interactivity between attributes.Under the linguistic term set, the experts employ the linguistic expressions generated by the context-free grammar to qualitatively describe the interactivity between attributes.2) Through the conversion function, the linguistic expressions are transformed into HFLTSs.The individual evaluation results of all experts are further aggregated with the defined hesitant fuzzy linguistic weighted power average operator (HFLWPA).3) Based on the standard Euclidean distance formula of the hesitant fuzzy linguistic elements (HFLEs), the hesitant fuzzy linguistic interaction degree (HFLID) between attributes is defined and calculated by constructing a piecewise function.As a result, a 2-order additive fuzzy measure identification method based on HFLID is proposed.
The advantages of this paper may be as follows: 1) The proposed method gives the definition of interactivity between attributes by using the supermodular game theory, which makes the interaction between attributes easy to explain and understand.2) According to this definition, a linguistic term set is established to characterize the interactivity between attributes.Under the linguistic term set, the proposed method allows the experts to qualitatively describe the interactivity between attributes by using linguistic expressions generated by the context-free grammar, when they are hesitant among multiple possible linguistic information, which reflects both fuzziness and hesitation of decision-making and makes the evaluation of the interactivity between attributes closer to reality.3) Through the conversion function, the linguistic expressions are transformed into the HFLTSs.The individual evaluation results of all experts are further aggregated by using the defined HFLWPA, which not only considers the weights of experts, but also considers the mutual support degree of opinions of experts, thereby ensuring the rationality of decision-making.4) The proposed method uses the standard Euclidean distance formula of HFLEs to define and calculate the HFLID between attributes, so the transformation from qualitative description to quantitative characterization is finally realized, thereby better preserving, and characterizing the experts' evaluation information.
This work proposed a 2-order additive fuzzy measure identification method based on HFLID.Obviously, on the one hand, compared with the objective methods describing and depicting the interactivity between attributes (as in 7,17-21 and 22 ), the proposed method has subjectivity.On the other hand, compared with the subjective methods describing and depicting the interactivity between attributes (as in 9-15 and 16 ), the proposed method can reflect both fuzziness and hesitation.www.nature.com/scientificreports/By using the HFLTS, this work successfully reflects both fuzziness and hesitation in evaluating the interactivity between attributes in the identification process of 2-order additive fuzzy measure.The proposed method can be widely applied to multi-attribute decision-making problems such as credit evaluation, ESG evaluation, competitiveness evaluation, and technological innovation capability evaluation, which has high practical application value and broad application prospects.
The number of samples and experts used in the application example (see "Application Example") is small, which weakens the persuasiveness of the analysis results and the stability of the interactivity evaluation.In the future, the number of samples and experts should be increased.Furthermore, the established linguistic term set S contains only nine linguistic terms, which is relatively extensive.Therefore, it is necessary to add the linguistic terms in S to improve the scientificity of decision-making in the future.At the same time, the credit assessment index system for big data enterprises (see Table 1 in 16 ) includes only six primary indicators, it is easier to calculate the 2-order additive fuzzy measure.For example, experts only need to evaluate the interactivity of C 2 6 = 15 pairs of attributes and the 2-order additive fuzzy measure only need to determine 6 × (6 + 1) 2 = 21 parameters.However, when there are many primary indicators, the amount and complexity of calculation will increase significantly.Therefore, it is necessary to design calculation programs to complete calculations in the future.In addition, Method 1 should be compared with other methods except Method 2, Method 3 and Method 4, such as the proportional scaling method 11 , multicriteria correlation preference information method 12 , qualitative crossimpact analysis method 15 , etc.This is one of the shortcomings of this work and needs to be improved in the future.

Figure 1 .
Figure 1.The identification process of 2-order additive fuzzy measure based on HFLID. σ

be a set of sub- scripts of attributes.
P(X)

Table 1 .
Individual evaluation results of interactivity.
Attributes C

Table 3 .
Calculation results of Möbius transformation coefficients.

Table 4 .
Calculation results of 2-order additive fuzzy measures.